A common situation in practical industrial applications related to product development is the need to perform quick surveys inside a space of state parameters. In mature and very competitive industrial sectors like aerospace, this need is motivated by the drive to generate products having good technical performance within design cycles that are as short as feasible. That is: time is a key factor in aerospace competitiveness because shortening the time market may provide a leading economic advantage during the product life cycle.
In the specific case of aeronautics, the prediction of the aerodynamic forces, and more generally skin surface values distributions, experimented by an aircraft is an important feature, in order to optimally design its structural components so that the weight of the structure is the minimum possible, but at the same time being able to withstand the expected aerodynamic forces.
Thanks to the increase of the use of the Computer Fluid Simulation Capability, nowadays, the determination of the aerodynamic forces on an aircraft is commonly done by solving numerically the Reynolds Averaged Navier-Stokes equations that model the movement of the flow around the aircraft, using discrete finite elements or finite volume models. With the demand of accuracy posed in the aeronautical industry, each one of these computations requires important computational resources.
The dimensioning aerodynamic forces are not known a priori, and since the global magnitude of the forces may depend on many different flight parameters, like angle of attack, angle of sideslip, Mach number, control surface deflection angle, it has been necessary to perform many lengthy and costly computations to properly predict the maximum aerodynamic forces experimented by the different aircraft components or the complete aircraft.
In order to reduce the overall number of these lengthy computations, approximate mathematical modelling techniques have been developed in the past, like Single Value Decomposition (SVD) as a mean to perform intelligent interpolation, or the more accurate Proper Orthogonal Decomposition (POD from now onwards) that takes into account the physics of the problem by using a Galerkin projection of the Navier-Stokes equations.
Given a set of N scalar flow fields in a scalar variable φ (as the pressure), calculated using Computational Fluid Dynamics (CFD), POD methodology provides N mutually orthogonal POD modes Φi({right arrow over (x)}). POD methodology also provides the singular values of the decomposition, which allows truncating the number of POD modes to n<N modes, where n can be selected with the condition that the manifold contains the reconstruction of all computations (also called snapshots hereinafter) within a predetermined error. The manifold spanned by these modes is known as POD manifold, and is the manifold that minimizes distance from the snapshots among the manifolds of dimension n.
POD modes allow us to reconstruct every snapshot as
      ϕ    ⁡          (                        x          →                ,        AoA        ,        M            )        =            ∑      i        ⁢                            a          i                ⁡                  (                      AoA            ,            M                    )                    ⁢                        Φ          i                ⁡                  (                      x            →                    )                    where the scalars ai are called POD-mode amplitudes and can be calculated upon orthogonal projection of the snapshot on the POD manifold. If the snapshots are appropriately selected, then the POD manifold contains a good approximation of the flow field for values of the parameters (such as angle of attack AoA and the Mach number M), in a given region of the parameter space. For general values of the parameters (not corresponding to the snapshots), we still expand the flow variables in terms of the POD modes as in the equation above and calculate the POD mode amplitudes using a Genetic Algorithm (GA), which selects the amplitudes as the minimizers of a properly defined residual of the governing equations and boundary conditions; such method will be called Genetic Algorithm+Proper Orthogonal Decomposition (GAPOD) hereinafter. This approach provides a good approximation to the exact solution and is flexible, robust, and fairly independent of the number and location of the CFD-calculated snapshots in the parameter space.
However, if the snapshots used to generate the POD manifold exhibit shock waves (as must be expected in transonic conditions) that move significantly as the parameters are varied, then either (a) the resulting POD modes are stair-like shaped (instead of exhibiting the correct one-jump, shock wave shape), which yields a poor approximation, or (b) both the number of required snapshots and the dimension of the POD manifold are quite large. This is because POD approximations consist of linear combinations, and linear combinations of shifted jumps do not give jumps but stairs. This fact implies that low Mach number flow configurations can be predicted with a few POD modes using a plain GAPOD methodology, but high Mach number cases require a more sophisticate method, to preserve the shock wave structures.
The present invention is intended to attend this demand.